A uniform result for the dimension of fractional Brownian motion level sets

Let $B=\{B_t:t\ge 0\}$ be a real-valued fractional Brownian motion of index $H\in (0,1)$. We prove that the macroscopic Hausdorff dimension of the level sets $$\mathcal{L}$x = \left{ t \in $\mathbb{R}$+ , : , B_t=x \right}$is,withprobabilityone,equalto$1-H$forall$x\in$\mathbb{R}$$.